About the notation of Axiom of Regularity

Question

Axiom of Regularity/Foundation

∀? (? ≠ ∅ → ∃? (? ∈ ? ∧ ? ⋂ ? = ∅))
This says that every non-empty set must contain a set that shares no
elements with it.

So, how do we understand axiom of regularity is talking about a set with "y" just from the notation?
(I think using intersection ⋂ gives a clue that y is a set because only sets intersects not elements but not sure if its right inference.)

Noah Schweber · Accepted Answer

There's less here than meets the eye. In $mathsf{ZFC}$, everything is a set. So $y$ is a set because it can't not be.
(This is the source of the most common criticism of $mathsf{ZFC}$, incidentally: that formalizing mathematics in $mathsf{ZFC}$ inevitably leads to "junk theorems." See here for example.)

About the notation of Axiom of Regularity

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